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Merge trees, a type of topological descriptor, serve to identify andsummarize the topological characteristics associated with scalar fields. Theypresent a great potential for the analysis and visualization of time-varyingdata. First, they give compressed and topology-preserving representations ofdata instances. Second, their comparisons provide a basis for studying therelations among data instances, such as their distributions, clusters,outliers, and periodicities. A number of comparative measures have beendeveloped for merge trees. However, these measures are often computationallyexpensive since they implicitly consider all possible correspondences betweencritical points of the merge trees. In this paper, we perform geometry-awarecomparisons of merge trees using labeled interleaving distances. The main ideais to decouple the computation of a comparative measure into two steps: alabeling step that generates a correspondence between the critical points oftwo merge trees, and a comparison step that computes distances between a pairof labeled merge trees by encoding them as matrices. We show that our approachis general, computationally efficient, and practically useful. Our generalframework makes it possible to integrate geometric information of the datadomain in the labeling process. At the same time, it reduces the computationalcomplexity since not all possible correspondences have to be considered. Wedemonstrate via experiments that such geometry-aware merge tree comparisonshelp to detect transitions, clusters, and periodicities of time-varyingdatasets, as well as to diagnose and highlight the topological changes betweenadjacent data instances.

Geometry-Aware Merge Tree Comparisons for Time-Varying Data with Interleaving Distances